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International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395-0056

Volume: 02 Issue: 06 | Sep-2015 www.irjet.net p-ISSN: 2395-0072

© 2015, IRJET ISO 9001:2008 Certified Journal 013

Analysis of Natural Frequencies for Cantilever Beam with I- and

T- Section Using Ansys Vipin Kumar1, Kapil Kumar Singh

2, Shwetanshu Gaurav3,

1 M.Tech Scholar,Department of Mechanical Engineering NITTTR, Chandigarh, India 2Department of Mechanical Engineering, Government Polytechnic Tundla (U.P), India

3 Department of Mechanical Engineering Meerut Institute of Technology, Meerut, India

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Abstract - Modal analysis is a process to determine the vibration characteristics (natural frequencies and mode shapes) of a structure or a machine component while it is being designed. It has become a major alternative to provide a helpful contribution in understanding control of many vibration phenomena which encountered in practice. In this work we compared the natural frequency for different material having same I and T cross- sectional beam. The cantilever beam is designed and analyzed in ANSYS. The cantilever beam which is fixed at one end is vibrated to obtain the natural frequency, mode shapes and deflection with different loads. Key Words: I-Section, T-Section, Mode Shapes, Natural Frequency

1. INTRODUCTION In engineering field vibration behavior of an element plays a key role without which it is incomplete. Resonance is a key aspect in dynamic analysis, which is the frequency of any system matches with the natural frequency of the system which may lead to catastrophes or system failure. Modal analysis has become a major alternative to provide a helpful contribution in understanding control of many vibration phenomena which encountered in practice [1]. A new finite element model for laminated composite beams. The model includes sufficient degrees of freedom to allow the cross-sections of each lamina to deform into a shape which includes up through cubic terms in thickness co-ordinate. The element consequently admits shear deformation up through quadratic terms for each lamina but not interfacial slip or delamination [2]. Higher order shear deformation theory is used for the analysis of composite beams. Nine nodes are parametric elements are used in the analysis. Natural frequencies of composite beam are compared for different stacking sequences, different (l/h) ratios and different boundary conditions. They had shown that natural frequency decreases with an increase in ply angle and a decrease in (l/h) ratio [3].The symbolic computation technique to analyze the free vibration of generally layered composite beam on the basis of a first-order shear deformation theory. The model

used considering the effect of poisson effect, coupled extensional, bending and torsional deformations as well as rotary inertia [4]. It has investigated the free vibration of axially laminated composite Timoshenko beams using dynamic stiffness matrix method. This is accomplished by developing an exact dynamic stiffness matrix of a composite beam with the effects of axial force, shear deformation and rotatory inertia taken into account. The effects of axial force, shear deformation and rotator inertia on the natural frequencies are demonstrated. The theory developed has applications to composite wings and helicopter blades [5]. A finite element model to investigate the natural frequencies and mode shapes of the laminated composite beams. The FE model needed all lamina had the same lateral displacement at a typical cross-section, but allowed each lamina to rotate to a different amount from the other. The transverse shear deformations were included [6]. The effects of the location and depth of the cracks, and the volume fraction and orientation of the fibers on the natural frequencies and mode shapes of the beam with transverse non-propagating open cracks, were explored. The results of the study leaded to conclusions that, presented method was adequate for the vibration analysis of cracked cantilever composite beams, and by using the drop in the natural frequencies and the change in the mode shapes, the presence and nature of cracks in a structure can be detected [7]. They had done free vibration analysis of a cross-ply laminated composite beam on Pasternak Foundation. The model is designed in such a way that it can be used for single-stepped cross section. For the first time to-date, the same analysis was conducted for a single-stepped LCB on Pasternak foundation. Stiffness and mass matrices of a cross-ply LCB on Pasternak foundation using the energy method are computed [9]. The cracks can be present in structures due to their limited fatigue strengths or due to the manufacturing processes. These cracks open for a part of the cycle and close when the vibration reverses its direction. These cracks will grow over time, as the load reversals continue, and may reach a point where they pose a threat to the integrity of the structure. As a result, all such structures must be carefully maintained and more generally, SHM denotes a reliable system with the ability



© 2015, IRJET ISO 9001:2008 Certified Journal Page 1014

to detect and interpret adverse “change” in a structure due to damage or normal operation. [10]. The finite beam element was formulated using the composite element method with a one-member–one-element configuration with cracks where the interaction effect between cracks in the same element was automatically included. The accuracy and convergence speed of the proposed model in computation were compared with existing models and experimental results. [11].The effects of crack depth and location, fiber orientation, and fiber volume fraction on the flexibility and consequently on natural frequency and mode shapes for cracked fiber-reinforced composite beams are investigated [12]. Vibration is a mechanical oscillation about a reference position. Any system has certain characteristics to be fulfilled before it will vibrate. To put in simple words, every system has a stable position in which all forces are equivalent and when this equilibrium is disturbed, the system will try to regain its stable position To remain stable, structure exhibits vibration at different magnitude when excited, the degree of vibration varies from point to point (node to node), due to the variation of dynamic responses of the structure and the external forces applied Therefore, vibration may also be described as the physical manifestation of the interchange between kinetic and potential energy [13]. The mechanical properties of aluminum and fiber (Nylon and Glass fiber reinforcement plastic) are measured a universal testing machine. The three-dimensional finite element models of composite beam with and without cracks are constructed and then computational modal analysis on ANSYS-14 is then performed to generate natural frequencies and mode shapes [14] They considered geometric non-linearity and have solved the expression using Variation Iteration Method (VIM). Also the different nonlinear frequencies have been considered for different shapes of modes [15].

2. MATHEMATICAL MODEL

In the present case I and T section length of 1500 mm long

and having the cross-sectional area of as shown in figure 1

is considered. One end of the beam is fixed and the system

is subjected to vibration. The material which is assumed is

structural steel, cast iron and stainless steel.

(a)

(b)

Figure 1:- Mathematical Model of (a) I-Section, (b) T-

Section




Beam Specifications:

Table 1:- Dimesion and Properties of a Cantilever beam

The dimension and properties of a cantilever beam with I

and T cross section has one end fixed and other end free as

shown in table-1, and load applied to free end.

3. GOVERNING EQUATION [14]

3.1 Modal Analysis

3.1.1 Damping Matrices

Damping may be introduced into a transient, harmonic, or damped modal analysis as well as a response spectrum. The type of damping allowed depends on the analysis as described in the subsequent sections.

3.1.2 Transient Analysis and Damped Modal Analysis

The damping matrix, [C], may be used in transient and damped modal analyses as well as substructure generation. In its most general form, the damping matrix is composed of the following components.

E

i j

1 1 1

1 1

2 2 1[M] ( g)[K] [M ] [( g g )[K ]] [C ]

1[C

[C]

] [G ]

ma m e

gv

N N Nm m

i j j j k

i j k

NN

m l

m l

(1) where: [C] = structure damping matrix, α = mass matrix multiplier, [M] = structure mass matrix, β = stiffness matrix multiplier, [K] = structure stiffness matrix, Nma = number of materials, αi

m = mass matrix multiplier for material i, [Mi] = portion of structure mass matrix based on material i, Nmb = number of materials, [Kj] = portion of structure stiffness matrix based on material j, Ne = number of elements with specified damping, [Ck] = element damping matrix, Ng = number of elements with Coriolis or gyroscopic damping, [Gl] = element Coriolis or gyroscopic damping matrix, βj

m = stiffness matrix multiplier for material.

3.1.3 Harmonic Analysis

The damping matrix ([C]) used in harmonic analyses is composed of the following components.

E

i j

1 1

1 1 1

2 2 1[M] ( g)[K] [M ] [( g g )[K ]]

1

[C

[C ] [C ] [G ]

]ma m

ge v

N Nm m

i j j j

i j

NN N

k m l

k m l

(2)

The input exciting frequency, Ω, is defined in the range between ΩB and ΩE via

ΩB = 2πfB , ΩE = 2πfE, fB = beginning frequency, fE = end frequency

Substituting equation (2) into the harmonic response equation of motion and rearranging terms yields

E

j j j m i

2

k 1 2 1

[[K] 2 [K] (2g g )[K ] [C ] [ [M] [M ] [K]

[K ] [C ] [G ]]](u ) F [M]

m

i

m

j j j

i g i

iu

(3)

The complex stiffness matrix in the first row of the equation consists of the normal stiffness matrix augmented by the structural damping terms given by g, gi, gi

E, and [Cm] which produce an imaginary contribution. Structural damping is independent of the forcing frequency, Ω, and produces a damping force proportional to displacement (or strain). The terms g, gi, and gi

E are damping ratios (i.e., the ratio between actual damping and critical damping, not to be confused with modal damping). The second row consists of the usual viscous damping terms and is linearly dependent on the forcing frequency, Ω, and produces forces proportional to velocity.

3.1.4 Mode-Superposition Analysis

The damping matrix is not explicitly computed, but rather the damping is defined directly in terms of a damping ratio ξd. The damping ratio is the ratio between actual damping

and critical damping. The damping ratio d

i for mode i is the combination of

2 2

d m

i i i

i

(4)

ξ = constant modal damping ratio, m

i = modal damping

ratio for mode shape i (see below), ωi = circular natural frequency associated with mode shape i = 2πfi , fi = natural frequency associated with mode shape i, α = mass matrix multiplier

The modal damping ratio m

i can be defined for each

mode directly (undamped modal analyses only). Alternatively, for the case where multiple materials are present whose damping ratios are different, an effective

Dimension/ Properties

Structural Steel

Grey Cast Iron

Stainless Steel

Length (mm) 1500 1500 1500

Width (mm) 100 100 100

Thickness (mm) 10 10 10

Height (mm) 200 200 200

Young Modulus (MPa) 2x105 1.1x105 1.93x105

Density( Kg/m3) 7850 7200 7750

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mode-dependent damping ratio m

i can be defined in the

modal analysis if material-dependent damping is defined and the element results are calculated. This effective damping ratio is computed from the ratio of the strain energy in each material in each mode using,

1

1

m

m

Nm s

j j

jm

i Ns

j

j

E

E

4. Result and Analysis

4.1 Natural frequency and deflection on various loads In order to analyze the natural frequency and deflection of I

and T cross sectional cantilever beam, first we calculate the

mechanical properties of structural steel, cast iron and

stainless steel. These mechanical properties are fed into

ANSYS-14 to calculate the deflection and natural frequency

for cantilever beam.

Table 2:- Deflection and Natural frequency of Structural

Steel for T- Section

T-Section Steel

Load (KN) Deflection(mm) Natural frequency

10 4.73 19.028

20 9.46 54.141

30 14.19 79.988

40 18.92 97.096

50 23.65 171.64

60 28.38 222.82

Table 3:- Deflection and Natural frequency of Cast Iron for

T-Section

T-Section Cast Iron


10 8.6 14.377

20 17.2 43.979

30 25.8 61.935

40 34.4 77.247

50 43 138.98

60 51.6 173.61

Table 4:- Deflection and Natural frequency of Stainless steel

for T-section

T-Section Stainless Steel


10 4.9 18.78

20 9.8 53.52

30 14.7 79.082

40 19.6 95.923

50 24.5 169.76

60 29.4 219.98

Table 5:- Deflection and Natural frequency of Structural

Steel for I- Section

I-Section Structural Steel

Load(KN) Deflection(mm) Natural frequency

10 2.59 26.341

20 5.18 56.716

30 7.77 94.068

40 10.36 161.14

50 12.95 230.46

60 15.54 415.08 Table 6:- Deflection and Natural frequency of Cast Iron for I-

Section

I-Section Cast Iron


10 4.751 20.372

20 9.5 44.045

30 14.25 72.857

40 19 124.75

50 23.75 178.88

60 28.5 321.16




Table 7:- Deflection and Natural frequency of Stainless steel

for I-section

I-Section Stainless Steel


10 2.687 26.043

20 5.374 55.939

30 8.061 92.994

40 10.74 159.32

50 13.43 227.59

60 16.12 410.54 As the load on the cantilever beam increases, the deflection

also increases. The deflection is minimum at 10N of I-section

for structural steel whereas deflection is maximum at 50 KN

of T-section for Cast iron as shown in table 2-7.

(a)

(b)

(c)

Figure 2:- Comparison between deflection for T and I

section (a) Structural Steel (b) cast Iron, (c) Stainless

Steel

(a)

(b)




(c)

Figure 3:- Comparison between natural frequency for T

and I section (a) Structural Steel (b) cast Iron, (c)

Stainless Steel

4.2 Natural frequency for different mode shapes for T-section

Figure 4:- 1

st mode of vibration for T-section

Figure 5:- 2

nd mode of vibration for T-section

Figure 6:- 3

rd mode of vibration for T-section

Figure 7:- 4th

mode of vibration for T-section

Figure 8:- 5

th mode of vibration for T-section




Figure 9:- 6th

mode of vibration for T-section

The first mode of vibration for T- section is a bending mode

in horizontal direction. In this mode shape, the frequency is

18.78 Hz. The beam is tending to bend about the root

section’s minimum moment of inertia. The second modes of

vibration is also bending mode having natural frequency 53.52

Hz. The third mode of vibration is also bending mode in

vertical direction. The frequency of the third mode shape is

79.08 Hz. The fourth mode of vibration is twisting about the

root, t he frequency is affected by tip rotational moment of

inertia. The frequency of the fifth mode is 95.92 Hz. The fifth

mode of vibration is bending and twisting mode in horizontal

with frequency 169.76 Hz. The sixth mode shape is twisting

with natural frequency 219.98 Hz

4.3 Natural frequency for different mode shapes for I-

section

Figure 10:- 1

st mode of vibration for I-section

Figure 11:- 2

nd mode of vibration for I-section

Figure 12:- 3

rd mode of vibration for I-section

Figure 13:- 4th

mode of vibration for I-section

Figure 14:- 5th


Figure 15:- 6th





The first mode of vibration for I- section is a bending mode in

horizontal direction. In this mode shape, the frequency is

26.34 Hz. The beam is tending to bend about the root

section’s minimum moment of inertia. The second modes of

vibration is also bending mode having natural frequency 56.71

Hz. The third mode of vibration is also bending mode in

vertical direction. The frequency of the third mode shape is

94.06 Hz. The fourth mode of vibration is twisting about the

root, t he frequency is affected by tip rotational moment of

inertia. The frequency of the fifth mode is 161.14 Hz. The

fifth mode of vibration is bending and twisting mode in

horizontal with frequency 230.46 Hz. The sixth mode shape is

twisting with natural frequency 415.08.

5. Conclusion: The following conclusions are as given below.

The deflection of I section is less than the T- section

Maximum deflection is seen in T-section for cast iron i.e. 51.6 mm for 50 KN load.

The Minimum deflection is seen in I-section structural steel i.e. 2.59 mm for 10 KN load.

The minimum natural frequency is obtained in T-section for cast iron i.e. 14.3 Hz, whereas the maximum natural frequency is obtained in I section for structural steel i.e. 415.08 Hz

REFERENCES [1] Chandradeep Kumar, Anjani Kumar Singh, Nitesh

Kumar, Ajit Kumar, "Model Analysis and Harmonic Analysis of Cantilever Beam by ANSYS" Global journal for research analysis, 2014, Volume-3, Issue-9, PP:51-55.

[2] Yuan, F.G. and R.E. Miller. A higher order finite element for laminated composite beams. Computers & Structures, 14 (1990): 125-150.

[3] Dipak Kr. Maiti& P. K. Sinha. Bending and free vibration analysis of shear deformable laminated composite beams by finite element method. Composite Structures, 29 (1994): 421- 431

[4] Teboub Y, Hajela P. Free vibration of generally layered composite beams using symbolic computations. Composite Structures, 33 (1995): 123–34.

[5] Banerjee, J.R. Free vibration of axially loaded composite Timoshenko beams using the dynamic stiffness matrix method. Computers & Structures, 69 (1998): 197-208

[6] Bassiouni AS, Gad-Elrab RM, Elmahdy TH. Dynamic analysis for laminated composite beams. Composite Structures, 44 (1999): 81–7.

[7] Kisa “Free vibration analysis of a cantilever composite beam with multiple cracks”. Composites Science and Technology 64, 1391–1402. 2003.

[8] Tita and Carvalho “Theoretical and experimental dynamic analysis of fiber reinforced composite beams”. Journal of the Brazilian society of Mechanical Sciences and Engineering.Vol. xxv, No.3. 2003.

[9] R.A. Jafari-Talookolaei and M.T.Ahmadian.Free Vibration Analysis of a Cross-Ply Laminated Composite Beam on Pasternak Foundation. Journal of Computer Science, 3 (2007): 51-56.

[10] Ramanamurthy .“Damage detection in composite beam using numerical modal analysis”.International Journal on Design and Manufacturing Technologies, Vol.2, No.1. 2008.

[11] Lu and Law “Dynamic condition assessment of a cracked beam with the composite element model”. Mechanical Systems and Signal Processing, 23, 415–431.2009.

[12] Ali and Aswan “Free vibration analysis and dynamic behavior for beams with cracks”. International Journal of science engineering and Technology, Vol.2, No. 2. 2009.

[13] Veletsos A. S. and Newmark N. M., “A method for calculating the natural frequencies of continuousbeams”, engineering studies, Structural research, series no. 38, University of Illinois, Urbana, January, 1953.

[14] Mehdi. H et al “Modal Analysis of Composite Beam Reinforced by Aluminium-Synthetic Fibers with and without Multiple Cracks Using ANSYS” International journal of Mechanical Engineering, vol-4 issue-2, pp 70-80, 2014

[15] M. Baghani, H. Mazaheri, H. Salarieh . “Analysis of large amplitude free vibrations of clamped tapered beams on a nonlinear elastic foundation”. Applied Mathematical Modelling. 38, 1 February 2014, 1176-1186.

[16] Mahmood. A, Khan A A, Mehdi. H, “Vibration Analysis of Beam Subjected To Moving Loads Using Finite Element Method” IOSR journal of Engineering, vol-4, issue-5, pp- 07-17, 2014.

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